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GAZA, Gaza Strip – Thousands of Palestinian residents of the northern Gaza Strip fled their homes on Sunday and sought safety in U.N. shelters, heeding warnings from the Israeli military about impending plans to bomb the area in the sixth day of an offensive against Hamas that has killed more than 160 people.

The fighting showed no signs of slowing, despite international calls for a cease-fire and growing concerns about the mounting civilian death toll in Gaza. U.S. Secretary of State John Kerry spoke to Israeli Prime Minister Benjamin Netanyahu and voiced U.S. “readiness” to help restore calm, while Egypt, a key mediator between Israel and Hamas, continued to work behind the scenes.

U.N. Secretary-General Ban Ki-moon called for an immediate cease-fire in a statement issued late Sunday by his spokesman’s office.

Ban “strongly believes that it is in the interest of both sides that steps toward dangerous escalation be replaced with immediate measures to end the fighting, thus preventing further casualties and greater risks to regional peace and security,” it said.

WATCH: Israeli Prime Minister Benjamin Netanyahu defended his country’s military action in Gaza Strip during two interviews with American media Sunday

Ban condemned Hamas’ indiscriminate firing of rockets against Israeli civilian targets as “a violation of international law,” it said. He abhorred “the image of Israeli families hovering in shelters in fear of their children’s safety” and demanded “an immediate cessation of these indecent attacks.”

At the same time, the U.N. chief is “deeply worried about the impact on Palestinian families of Israeli military action. Too many Palestinian civilians have been killed, and any Israeli ground offensive will undoubtedly increase the death toll and exacerbate civilian suffering in the Gaza Strip,” it said.

Ban noted that despite the U.N. Security Council’s demand for a cease-fire, “the situation in and around the Gaza Strip appears to be worsening,” it said.

Amid the diplomacy, Israel said it was pushing forward with preparations for a possible ground invasion of Gaza. Thousands of troops have massed along the border in recent days.

“We don’t know when the operation will end,” Israeli Prime Minister Benjamin Netanyahu told his Cabinet on Sunday. “It might take a long time.” He said the military was prepared “for all possibilities.”

Israel launched the offensive last Tuesday in what it said was a response to heavy rocket fire out of Hamas-controlled Gaza. The military says it has launched more than 1,300 airstrikes, while Palestinian militants have launched more than 800 rockets at Israel. The Palestinian Health Ministry in Gaza says 166 people have been killed, including dozens of civilians. There have been no Israeli fatalities, though several people have been wounded, including a teenage boy who was seriously injured by rocket shrapnel Sunday.

Early Sunday, the Israeli air force dropped leaflets around the northern Gaza town of Beit Lahia ordering people to evacuate their homes. Israel says much of the rocket fire has come from the area, and overnight Sunday, the military carried out a brief ground operation on what it said was a rocket-launching site that could not be struck from the air. Four Israeli soldiers were lightly wounded before returning to Israel.

The U.N. refugee agency for Palestinians, UNRWA, said some 17,000 Palestinians had headed to special shelters set up in 20 United Nations schools in Gaza.

WATCH: The U.N. security council today called for a ceasefire between Israel and Hamas but as Mike Drolet reports, that call is so far falling on deaf ears

“The fact that in a span of almost a few hours, 10,000 people sought refuge in these 15 schools is an indication to the difficult situation on the ground,” said Sami Mshasha, a UNRWA spokesman.

Some raced by in pickup trucks, waving white flags. “Once we received the message, we felt scared to stay in our homes. We want to leave,” said one resident, Mohammed Abu Halemah.

Shortly before nightfall, Israel carried out a series of airstrikes in the northern Gaza town of Beit Lahia. Hamas’ Al-Aqsa TV station reported four airstrikes in a 10-minute span, and a large plume of black smoke could be seen over the area from the Israeli border. There were no immediate reports of casualties.

Hamas, an Islamic militant group sworn to Israel’s destruction, has remained defiant, and it continued to fire rockets into Israel throughout the day. It urged people in northern Gaza to stay in their homes and has so far rejected proposals for a cease-fire as unsatisfactory.

“They want us to put down our arms and leave the resistance,” said Moussa Abu Marzouk, a top Hamas official, on his Facebook page. “They started the battle, and we will stay on our land and fight to protect our future.”

Despite Israeli claims that it has inflicted heavy damage on the group, Hamas says it is largely unscathed, and Palestinian medics say most of the dead have been civilians.

IN PHOTOS: Israel’s military offensive targets Gaza Strip

A Palestinian searches the rubble of a center for disabled, that is littered with posters soliciting donations to support the Al Mabara association for handicapped run by Hamas, after an Israeli missile strike in Beit Lahiya, in the northern Gaza Strip, Saturday, July 12, 2014.

Adel Hana/AP Photo

A Palestinian firefighter hoses a boat hit in an missile strike at the port in Gaza City, Friday, July 11, 2014. Israel launched the Gaza offensive to stop incessant rocket fire that erupted after three Israeli teenagers were kidnapped and killed in the West Bank and a Palestinian teenager was abducted and burned to death in an apparent reprisal attack. The military says it has hit more than 1,100 targets already, mostly what it identified as rocket-launching sites, bombarding the territory on average every five minutes. (AP Photo/Hatem Moussa)

A Palestinian searches the ruins of the Al-Tawfeeq mosque after it was hit by an Israeli missile strike in the Nuseirat refugee camp, central Gaza Strip, Saturday, July 12, 2014.

Hatem Moussa/AP Photo

Palestinian medics place an injured man on a stretcher in the Shifa hospital after an Israeli missile strike in Gaza City on Saturday, July 12, 2014. Palestinian officials said Israeli airstrike hit a group of people sitting in front of a closed shop killing several, raising the Palestinian death toll to more than 120 during an Israeli offensive that showed no signs of slowing down. (AP Photo/Khalil Hamra)

Palestinians inspect the rubble of a building after it was hit by an Israeli missile strike in Gaza City, Friday, July 11, 2014. Israel launched the Gaza offensive to stop incessant rocket fire that erupted after three Israeli teenagers were kidnapped and killed in the West Bank and a Palestinian teenager was abducted and burned to death in an apparent reprisal attack. The military says it has hit more than 1,100 targets already, mostly what it identified as rocket-launching sites, bombarding the territory on average every five minutes.(AP Photo/Hatem Moussa)

A firefighter stands at a gas station that was hit by a rocket fired from the Gaza Strip in the city of Ashdod, Israel, Friday, July 11, 2014. The attack on the gas station in Ashdod looked to be the most serious attack in Israel in the four days of fighting that has seen Israel deliver a heavy blow to Gaza’s Hamas leaders. The military have carried out more than 1,000 strikes against Gaza targets that have killed at least 95 people, including dozens of civilians.(AP Photo/Tsafrir Abayov)

Smoke rises following Israeli strikes on Gaza, seen from Israel-Gaza Border, Thursday, July 10, 2014. Israel dramatically escalated its aerial assault in Gaza Thursday hitting hundreds of Hamas targets, as Palestinians reported more than a dozen of people killed in strikes that hit a home and a beachside cafe and Israel’s missile defense system once again intercepted rockets fired by militants at the country’s heartland. (AP Photo/Lefteris Pitarakis)

An Israeli border police officer checks documents of a Palestinian man on his way to pray at the Al-Aqsa Mosque in Jerusalem during the Muslim holy month of Ramadan at the Qalandia checkpoint, between the West Bank city of Ramallah and Jerusalem, Friday, July 11, 2014. (AP Photo/Majdi Mohammed)

Palestinians search in the rubble of a destroyed house where eight members of the Al Haj family were killed in a strike early morning in Khan Younis refugee camp, southern Gaza Strip on Thursday, July 10, 2014. Israel dramatically escalated its aerial assault in Gaza Thursday hitting hundreds of Hamas targets, as Palestinians reported more than a dozen of people killed in strikes that hit a home and a beachside cafe and Israel’s missile defense system once again intercepted rockets fired by militants at the country’s heartland. (AP Photo/Khalil Hamra)

Palestinian mourners chant slogans as they carry the bodies of eight members of the Al Haj family, who were killed in an Israeli missile strike early morning, during their funeral in Khan Younis refugee camp, southern Gaza Strip on Thursday, July 10, 2014. Israel dramatically escalated its aerial assault in Gaza Thursday hitting hundreds of Hamas targets, as Palestinians reported more than a dozen of people killed in strikes that hit a home and a beachside cafe and Israel’s missile defense system once again intercepted rockets fired by militants at the country’s heartland. (AP Photo/Khalil Hamra)

The outbreak of violence follows the kidnappings and killings of three Israeli teenagers in the West Bank, the kidnapping and killing of a Palestinian teenager in an apparent revenge attack, and wide-ranging Israeli moves against Hamas militants and infrastructure in the West Bank. Hamas has demanded that hundreds of recently arrested activists be freed as part of a cease-fire.

Many of the airstrikes have been on the homes of wanted Hamas militants, putting their families at risk. In an attack on Saturday, the target of one such airstrike, Gaza’s police chief, survived, while 17 members of his extended family were killed.

Israel accuses Hamas of using Gaza’s civilians as human shields, putting people in the densely populated territory in danger.

“The leadership of Hamas and the other organizations has chosen – at a time when they are using the population of Gaza as human shields – to hide underground, to flee abroad and to deliberately put civilians in the line of fire,” Netanyahu said.

Despite Israel’s claims, the international community, including many of Israel’s allies, have begun to express concerns about the growing civilian death toll.

The Israeli military said that one of the rockets fired by Gaza militants Sunday night “hit an electricity infrastructure in Israel that supplied electricity to the Gaza Strip, causing a power outage to some 70,000 Gaza civilians.”

In Vienna, Kerry spoke Sunday with Netanyahu and highlighted U.S. concerns about the “escalating tensions,” the State Department said.

Kerry “described his engagement with leaders in the region to help to stop the rocket fire so calm can be restored and civilian casualties prevented, and underscored the United States’ readiness to facilitate a cessation of hostilities,” the State Department said.

VIDEO GALLERY:

UN security council President calls for ceasefire between Israel and Gaza

Egypt, meanwhile, said President Abdel Fattah el-Sissi spoke to the U.N. chief Ban Ki-moon. El-Sissi’s spokesman quoted Ban as praising Egyptian efforts to halt the fighting and affirming that “Egypt is the most capable party to effectively participate in reaching a calm between the two sides.” Netanyahu’s office declined comment on diplomatic efforts.

Other countries were also involved. Germany’s foreign minister said he would head to the region on Monday, while French President Francois Hollande tried to rally Arab and Muslim leaders to push for a cease-fire.

Hollande held telephone talks over the weekend with Turkish Prime Minister Recep Tayyip Erdogan and Tunisian President Moncef Marzouki.

Marzouki spokesman Adnane Mancer said the French and Tunisian presidents agreed that Marzouki would try to talk to Hamas leaders and urge a cease-fire, while Hollande would try to do the same with other parties. A French presidential official said Hollande was talking to Israeli, Palestinian and other Arab officials.

On Sunday, Palestinians with foreign passports began leaving Gaza through the Erez border crossing. Israel, which co-operated in the evacuation, said 800 Palestinians living in Gaza have passports from countries including Australia, Britain and the U.S.

Rawan Mohanna, a 21-year-old chemistry major at the University of Texas, said she had arrived in Gaza with her family a month ago because her older sister was getting married to a Gazan.

Mohanna, who lives in Dallas, said her family is now returning to the U.S. with mixed feelings because her newlywed sister and other relatives were staying behind.

“It’s bittersweet that we get to leave but they are still there and they can’t get out,” she said.

On Sunday night, Israel’s military said rockets were fired at Israel from both Syria and Lebanon in separate incidents. There were no injuries or damage, but Israel fears militant groups along its northern frontier may try to open a second front. The rocket attacks were the second such barrage on Israel from its northern neighbours in recent days.

The Israeli military said it retaliated by shooting toward the source of fire.

In Egypt, security officials said they had foiled a new attempt to fire rockets at Israel by militants in the northern Sinai Peninsula which borders Israel.